🔮Forecasting Unit 4 – Regression Analysis for Forecasting

What's Regression Analysis?

• Statistical technique used to model and analyze the relationship between a dependent variable and one or more independent variables
• Helps understand how changes in independent variables are associated with changes in the dependent variable
• Useful for making predictions or forecasts based on the relationships identified in the model
• Can be used to identify which independent variables have the most significant impact on the dependent variable
• Provides a measure of how well the model fits the data using metrics like R-squared and adjusted R-squared
• Allows for hypothesis testing to determine if the relationships between variables are statistically significant
• Enables the identification of outliers or influential observations that may impact the model's accuracy

Key Concepts in Regression

• Dependent variable (response variable) is the variable being predicted or explained by the model
• Independent variables (predictor variables) are the variables used to predict or explain the dependent variable
• Coefficient estimates represent the change in the dependent variable associated with a one-unit change in an independent variable, holding other variables constant
• P-values indicate the statistical significance of each independent variable in the model
• Confidence intervals provide a range of values within which the true population parameter is likely to fall
• Residuals represent the differences between the observed values of the dependent variable and the values predicted by the model
• Multicollinearity occurs when independent variables are highly correlated with each other, which can affect the interpretation of the model
• Interaction terms allow for the modeling of relationships where the effect of one independent variable depends on the value of another independent variable

Types of Regression Models

• Simple linear regression involves one independent variable and one dependent variable
  • Equation: $y = \beta_0 + \beta_1x + \varepsilon$
• Multiple linear regression involves multiple independent variables and one dependent variable
  • Equation: $y = \beta_0 + \beta_1x_1 + \beta_2x_2 + ... + \beta_kx_k + \varepsilon$
• Polynomial regression includes higher-order terms (squared, cubed, etc.) of the independent variables to capture non-linear relationships
• Stepwise regression iteratively adds or removes independent variables based on their statistical significance to find the optimal model
• Ridge regression and Lasso regression are regularization techniques used to handle multicollinearity and improve model performance
• Logistic regression is used when the dependent variable is binary or categorical
• Time series regression models account for the temporal dependence of data points (autoregressive models, moving average models, ARIMA models)

Building a Regression Model

• Define the problem and identify the dependent and independent variables
• Collect and preprocess data, handling missing values, outliers, and transformations if necessary
• Split the data into training and testing sets for model validation
• Select the appropriate regression model based on the nature of the problem and the relationships between variables
• Estimate the model coefficients using a method like Ordinary Least Squares (OLS) or Maximum Likelihood Estimation (MLE)
• Assess the model's goodness of fit using metrics such as R-squared, adjusted R-squared, and root mean squared error (RMSE)
• Refine the model by adding or removing variables, considering interaction terms, or applying regularization techniques
• Validate the model using the testing set to ensure its performance on unseen data

Assumptions and Diagnostics

• Linearity assumes a linear relationship between the dependent variable and independent variables
  • Check using scatter plots or residual plots
• Independence of observations assumes that the residuals are not correlated with each other
  • Check using the Durbin-Watson test or by plotting residuals against the order of observations
• Homoscedasticity assumes that the variance of the residuals is constant across all levels of the independent variables
  • Check using a scatter plot of residuals against predicted values or the Breusch-Pagan test
• Normality assumes that the residuals are normally distributed
  • Check using a histogram, Q-Q plot, or the Shapiro-Wilk test
• No multicollinearity assumes that the independent variables are not highly correlated with each other
  • Check using the correlation matrix or Variance Inflation Factor (VIF)
• Influential observations and outliers can significantly impact the model's coefficients and should be identified and addressed
  • Check using Cook's distance, leverage values, or standardized residuals

Interpreting Regression Results

• Coefficient estimates represent the change in the dependent variable associated with a one-unit change in the corresponding independent variable, holding other variables constant
• P-values indicate the statistical significance of each independent variable
  • A p-value less than the chosen significance level (e.g., 0.05) suggests that the variable has a significant impact on the dependent variable
• Confidence intervals provide a range of plausible values for the population parameters
  • Narrower intervals indicate more precise estimates
• R-squared measures the proportion of variance in the dependent variable explained by the independent variables
  • Values range from 0 to 1, with higher values indicating a better fit
• Adjusted R-squared accounts for the number of independent variables in the model and penalizes the addition of irrelevant variables
• Residual plots can help identify patterns or deviations from the assumptions of linearity, homoscedasticity, and normality

Forecasting with Regression

• Use the estimated regression model to make predictions or forecasts for new or future observations
• Input the values of the independent variables for the new observation into the regression equation to obtain the predicted value of the dependent variable
• Consider the uncertainty associated with the predictions by calculating prediction intervals
  • Prediction intervals account for both the uncertainty in the model coefficients and the inherent variability in the data
• Assess the accuracy of the forecasts using metrics such as Mean Absolute Error (MAE), Mean Squared Error (MSE), or Mean Absolute Percentage Error (MAPE)
• Update the model as new data becomes available to improve its forecasting performance
• Be cautious when extrapolating beyond the range of the data used to build the model, as the relationships may not hold outside the observed range

Limitations and Pitfalls

• Omitted variable bias occurs when important variables are not included in the model, leading to biased coefficient estimates
• Reverse causality can occur when the dependent variable influences one or more of the independent variables, violating the assumption of exogeneity
• Overfitting happens when the model is too complex and fits the noise in the data rather than the underlying relationships
  • Regularization techniques like Ridge and Lasso regression can help mitigate overfitting
• Underfitting occurs when the model is too simple and fails to capture important relationships in the data
• Outliers and influential observations can significantly impact the model's coefficients and should be carefully examined and addressed
• Multicollinearity can make it difficult to interpret the individual effects of the independent variables and lead to unstable coefficient estimates
• Autocorrelation in the residuals violates the assumption of independence and can lead to inefficient coefficient estimates and invalid inference
• Non-linear relationships may not be adequately captured by linear regression models, requiring the use of non-linear transformations or alternative modeling techniques